Theorem iccss 11595

Description: Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 20-Feb-2015.)

Assertion

Ref	Expression
iccss	|- ( ( ( A e. RR /\ B e. RR ) /\ ( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) )

Proof of Theorem iccss

Dummy variables x w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexr 9628	. . 3 |- ( A e. RR -> A e. RR* )
2		rexr 9628	. . 3 |- ( B e. RR -> B e. RR* )
3	1, 2	anim12i 564	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A e. RR* /\ B e. RR* ) )
4		df-icc 11539	. . 3 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
5		xrletr 11364	. . 3 |- ( ( A e. RR* /\ C e. RR* /\ w e. RR* ) -> ( ( A <_ C /\ C <_ w ) -> A <_ w ) )
6		xrletr 11364	. . 3 |- ( ( w e. RR* /\ D e. RR* /\ B e. RR* ) -> ( ( w <_ D /\ D <_ B ) -> w <_ B ) )
7	4, 4, 5, 6	ixxss12 11552	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) )
8	3, 7	sylan 469	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) )

Syntax hints:   -> wi 4   /\ wa 367   e. wcel 1823   C_ wss 3461   class class class wbr 4439  (class class class)co 6270   RRcr 9480   RR*cxr 9616   <_ cle 9618   [,]cicc 11535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-icc 11539

This theorem is referenced by:  xrhmeo  21612  lebnumii  21632  pcoval1  21679  pcoval2  21682  ivthicc  22036  dyaddisjlem  22170  volsup2  22180  volcn  22181  mbfi1fseqlem5  22292  dvcvx  22587  dvfsumle  22588  dvfsumabs  22590  harmonicbnd3  23535  ppisval  23575  chtwordi  23628  ppiwordi  23634  chpub  23693  cvmliftlem2  28995  fourierdlem76  32204  fourierdlem103  32231  fourierdlem104  32232  fourierdlem107  32235  fourierdlem112  32240